Session 2 Notes – reinforcement & reflection
Gavin Bell
We discussed terminology fist – everyone seemed reasonably happy with the technical and musical
terms after some discussion. Note that you don’t have always to explain terms such as Fourier
component mathematically (though equations are fine, of course!) – what’s important is that you
understand the concepts and can apply them to real music / musical sounds.
What is musical sound? Pressure waves in the air, as far as we’re concerned. This obviously
underpins everything we’ll discuss. Waves depend on the ability of a medium to oscillate; oscillations
depend on the medium’s elasticity and intertia. We developed these ideas from Hooke’s Law (linear
elasticity) through sound sources (oscillating speaker cones, vibrating air columns, etc.) to sound
waves in air. We briefly looked at the effects of temperature and humidity on the speed of sound in
air. You should know 𝑐 = 𝑓𝜆 and its consequences, e.g. that lower frequencies are associated with
longer wavelengths. Efficient sound sources cannot be much smaller than their associated
wavelengths, so cellos are bigger than violins, woofers are bigger than tweeters, and so on (actually
speakers are more complicated than this and depend strongly on their cabinet designs for bass
projection). We’ll look in more detail at oscillators in the Acoustic Instruments session.
We covered a few properties of sound, I played some very simple synthesised tones (made with
MATLAB, not a synthesiser, M-file “miniSynth.m”) and we measured some spectrograms.
The key idea is that a “musical” (non-tedious) sound contains many different frequencies. For a long,
steady sound, Fourier’s theorem tells us that the frequencies are all a multiple of some fundamental
f (2f, 3f, 4f, etc.). Octaves are associated with a doubling of frequency – we will look at pitch much
more carefully in session 3. You should have a reasonable idea of what frequencies are associated
with bass, middle and treble musical pitches (I always remember A4 = 440 Hz).
An important point from the synthesis is that the visual wave form (i.e. the pattern of amplitude as a
function of time) is NOT very informative as to the nature of the sound (its frequency content or
timbre). It is quite obvious to the ear, however, when we mix in extra harmonics and then
inharmonic sounds, as we demonstrated. The visual representation which is informative is the
frequency domain picture, i.e. a graph of power (“amount of sound”) vs. frequency. A spectrogram is
a representation of all the frequencies making up a sound as a function of time, so it’s a 3D graph
with axes of time and frequency plus a colour scale being the third “axis” and representing power.
The spectrograms shown overleaf have time on the horizontal axis (the clips are a few seconds long)
and frequency on the vertical axis (typically up to about 8 kHz) – this is the normal view in Audacity,
or indeed most audio analysis. It’s free software, and you can also get spectro-analysers for
phones/tablets so why not try looking at the frequency content of some other music?
Timbre is getting us closer to musical terminology – we describe timbres as “rich”, “harsh”, “tinny”,
“mellow” and so on. They reflect the particular balance of harmonics independent of loudness and
fundamental frequency.
But the real sound of a musical instrument is always characterised by both the frequency content
and the time variation. The sound of a single note is often described, in the time domain, by an
envelope, characterised by attack, decay, sustain and release times/amplitudes. Frequency content
changes with time as well, however, and it is the combination of time-domain and frequencydomain information which characterises a sound.
Violin spectrogram, raising pitch then finishing on artificial harmonic. Rich harmonic content except
for very pure final tone (black arrow).
<<violin.wav>>
Bass spectrogram - strongest below 500Hz (green arrow) as expected for bass, but lots of mid/high
as the notes are struck (yellow arrows) due to the strong transient sounds. Remember, Fourier
analysis tells us that a rapid change in the waveform (e.g. the sharp corner of a sawtooth wave or
the rapid hit of a drum) MUST be represented by a lot of high frequency power (low frequency
waves just don’t wobble rapidly enough).
<<bass.wav>>
Three recorders spectrogram with G being blown increasingly hard. Starts off with nice harmonic
series, goes a little sharp then register shifts up in two stages (onsets shown by arrows).
<<recorderhell.wav>>
Finally, loudness is a basic parameter. This useful table (from Canadian govt. H&S) shows some nonmusical sound sources. We use a log scale because of the high range of sound pressures to which the
ear can respond (a factor of a million: 20 × log10(1,000,000) = 120, the range of the scale).
Here is a useful page on dB, dB(A) scales: http://www.animations.physics.unsw.edu.au/jw/dB.htm
Make sure you understand the basics of these scales. The dB(A) scale accounts for the fact that the
human ear is more sensitive to frequencies 1 – 4 kHz which therefore sound louder.
Simple Excel exercise: try looking at decibels.xslx and plotting graphs of raw SPL and dB scale. You
can also check the formula for the dB scale.